Abstract
The McMillan-Mayer theory of multicomponent solutions is utilized to formulate a statistical-thermodynamic description of surfactant solution behavior from which quantitative predictions of micelle formation, micellar size distribution, and micellar solution phase separation can be made. Specifically, a model is constructed for the Gibbs free energy of the micellar solution, which is divided into ideal and excess contributions. The advantage of this approach is that it enables a systematic analysis of various models of intermicellar interactions. In this paper, we focus on micelles of nonionic surfactants which exhibit both repulsive and attractive interactions. The repulsive interactions are described using excluded-volume considerations, while the attractive ones are modeled using a mean-field description. Utilizing this statistical-thermodynamic framework, expressions for the chemical potentials of each of the solution components are obtained and used, along with the principle of multiple chemical equilibrium, to calculate the micellar size distribution and its moments. An analysis of the effect of excluded-volume interactions on the monomer and micelle concentrations and on the weight-average aggregation number of micelles which exhibit one-dimensional (cylindrical) growth indicates that these steric interactions promote micelle formation and growth. Interestingly, in the limit of extensive cylindrical micellar growth, we recover the well-known expressions for the micellar size distribution and its moments corresponding to the popular phenomenological ''ladder model'', with modified ''ladder model'' parameters which are explicit functions of the excluded-volume parameters. In addition, quantitative predictions of the critical micellar concentration, the polydispersity of the micellar size distribution, and phase separation characteristics are presented and found to compare favorably with available experimental data for aqueous micellar solutions of alkyl poly(ethylene oxide) nonionic surfactants.
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